How can I make a desert with sand that interacts with a car?
I have seen your Question and the video you have provided. Your first question is that whether it is possible to create that kickup effect. The answer is yes, but not the way you thought. You can't actually use a particle system for this one. For this, you need to use a smoke sim. They are easier to control and are way better. But I am warning you, YOU_ARE_GONNA_HAVE_A__BAD_TIME.
As for the tire tracks, that would probably need the particle system or some sort of addon like the molecular script. To create the dust simulation, you have to do this. Delete the default cube because its only purpose was to get deleted. Next, add a circle and flip it to be straight like this:
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Next, go to the particles tab and create a new particle system. Go to End and set it to 1 and set the lifetime to 3. Next, go to velocity and set the speed to something like 10 m/s or something you prefer, and set randomize to 3 or something higher than 1. Next, use quick smoke and it has turned into a quick smoke simulation. Do you think I am done? No, I am just getting started.
Now, click on the emitter, click on the physics tab and select the flow source as the particle system. Next, select initial velocity. Now click on the box thingy and go to settings and set the resolution division to 120 or higher or lower depending on the quality you want. Set the time scale to the minimum value as a dust sim is a slow sim.  Now as you want a dust sim, it is pretty clear that you want the dust to rise, but no so much, so set density to 0.2 in behavior. Next, skip to Vorticity and set it to 4, the maximum value, as dust swirls a lot.
If you want ultra high resolution, select a high resolution for a very high-quality dust sim.
For more realism, add a turbulence force field behind the emitter (the circle) and play with the noise for how much strength you want. I recommend going very high because dust goes very long. Now click on the black box and select dissolve and set time to a very high number because dust takes a lot of time to dissolve. Select adaptive domain. Next name your simulation cache whatever you want and bake the simulation if you are happy with your sim. Next,
go to shading and add a volume info node. There, pick density and attach it to density in the principled bsdf density. If you want more naturalism, add a math node and set it to multiply, and set the value to some large number like 30 or anything depending on the visibility of the dust. Keep in mind the math node should be in the middle of the volume info node and principled bsdf node. Set the color to whatever you like but I would recommend setting it to the hex code #C0A080, the literal color of a dust storm.
Next, you have to render this out obviously so you can render it on your computer if it is beefy, watch it burn if it is weak, or what I suggest, send it to a render farm like fox render. Please don't strike me for spam as I am just suggesting a name. You have your dust sim ready. As for the tire tracks, I would say use an addon, as the molecular script addon. Please note that you don't necessarily have to use the values in this answer, you can mess with the values as you prefer. I will be sharing a video link. It is mostly for reference and I would recommend going with the answer than the video. (Please don't strike me for a huge wall of text.)
Here's the link:
https://www.youtube.com/watch?v=sNHIdcjhJT4
Hope it helps.